Quantum Mechanics (1) · Problem 5.4

Rotations and angular momentum

Physics — Quantum Mechanics · su2-o3-homomorphism

Problem

Complete the proof of the following theorem: There is a two-to-one correspondence between

SU(2)

and

O^+(3)

under the mapping

u \in SU(2) \to R(u) \in O^+(3), \quad \text{where} \quad R_{ij}(u) = \frac{1}{2}\mathrm{Tr}\left(u^\dagger \sigma_i u \sigma_j\right), \tag{5.68}

and the rotation

R_{\mathbf{e}}(\theta)

corresponds to the pair

R_{\mathbf{e}}(\theta) \leftrightarrow \{u_{\mathbf{e}}(\theta), -u_{\mathbf{e}}(\theta) = u_{\mathbf{e}}(\theta + 2\pi)\}

. In particular, prove that the pair of elements

\{I, -I\} \subset SU(2)

maps to

I \in O^+(3)

, and that this mapping is a homomorphism:

R(u'u'') = R(u')R(u'')

for all

u', u'' \in SU(2)